1
$\begingroup$

Suppose $S$ be a closed orientable genous $g$ surface. Let $f$,$g$ be homeomorphis from $S$ to itself. Assume they induce the same map on 1st homology $H_1(S, \mathbb Z).$

My question is; does this imply $f$, $g$ are homotopic or isotopic?

Any help is appreciated. Thank you in advance.

$\endgroup$
2
  • 6
    $\begingroup$ No -- a Dehn twist about a nontrivial separating simple closed curve acts as the identity on homology, but is not nullhomotopic. The keyword to search for is "Torelli group". $\endgroup$ Jun 13, 2012 at 20:45
  • 3
    $\begingroup$ A thing to notice about your question is that homotopic homeomorphisms are automatically isotopic. $\endgroup$ Jun 13, 2012 at 21:46

1 Answer 1

8
$\begingroup$

Andy Putman's comment (Dehn twist on null-homologous curve; google for "Torelli group") is a concise and fairly complete answer to this question. I'm posting this CW answer so that the question does not resurface later.

$\endgroup$
2
  • 2
    $\begingroup$ Wouldn't you prefer a simplicial answer? (What a stupid joke!) $\endgroup$ Jun 14, 2012 at 6:19
  • 2
    $\begingroup$ Perhaps -- the possibilities for the answer were manifold. (Another bad joke.) $\endgroup$ Jun 14, 2012 at 13:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.